Mathematical modelling for rudder roll stabilization

MATHEMATICAL MODELLING FOR RUDDER ROLL STADILIZATION

by

J. van Amerongen

and J.C. van Cappelle

Control Laboratory

Electrical Engineering Dept.

Delft University of Technology

Postbox 5031, 2600 CA Delft,

The Netherlands

ABSTRACT

i:.dern

passenger

ships

as

well

as

naval

ships

are

equipped

with

roll

stabilization systems in order

to improve the passenger's

comfort or to keep

the

ship fully operational in bad weather conditions. Fins and tanks are most

commonly

used but both heve

disadvantages. Tanks require a

lot of space, fins introduce

a

considerable

drag

and are

expensive. Besides,

fin motions

disturb the

beading

control system, while rudder motions not only effect a ship's heading but influence

the

rolling motions

as well.

In present

systems this

interaction is

generally

disregarded. However, by explicitly

modelling the interaction it can

purposefully

be used by applying the rudder for roll stabilization as well. This paper describes

a siniplc' mathematical model for the

transfer between the rudder angle and the

two

outputs: rate of turn and roll angle. The parameters of this model can be estimated

from

full-scale

trials

such as

zig-zag tnanoeuvres.

Examples are

given of

the

parameter estimation of two different ships, a pilot vessel and a naval ship.

INTRODUCTION

Since a long time ago automatic control systems have been applied to controlling

the motions of a ship. In most

cases an autopilot for controlling the heading

has

replaced t.le

helmsman, although

manual steering remains

posiblc. To reduce

the

rolling

jnotic.us,

tanks

and

fins

have

been

applied

which

always work

fully

automatically. Until recently the controller structure of these systems was simple.

But the

availability of small

and inexpensive digital

computer systems offers

a

possibility

to

apply

more

advanced

control

algorithms

into a

wide range

of

practical systems. This has already led to a series of new autopilot designs, which

all

claim

more

accurate

and more

economical control

of a ship's heading,

by

introducing adaptive properties into the controller (see for instance Van Amerongen

and Van Nauta Lenke, 1978; Van Arnerongen

,

1981).

Although an autopilot which generates only smooth rudder motions implicitly

causes

less roll, this is seldom explicitly used as a design criterion. On the other hand,

the coupling between the stabilizer fins

and yawing are disregarded i.n the

design

as

veil.

To

get

an

optimal

performance

of both

systems the

ship should

be

considered

as one

multi-variable system

with two

inputs: rudder

angle and

fin

angle, and two outputs: heading and ruli angle; one integrated controller should be

designed for both actuators.

1ab v -Scheepsouwkuo

Technische Hogeschool

Another possibility with promising properties is to use the rudder not only for control of the heading but for roll stabilization as well (Carley, 1975; Cowley and Larobert, 1972, 1975; Lloyd, 1975, Baitis, 1980 ). Although this will require a more powerful steering machine the savings realized by not installing stabilizer fins are apparent. Also with respect to fuel economy a rudder roll stabilization (RRS) system may be advantageous. This aspect is of growing importar.ce. For merchant ships the total operational cost is already for more than sixty percent determined by the fuel cost. ( See figure 1 , according to MUch, 1980.)

13

29

lj

I

-I

fuel cost 1967 1976 1979

Figure 1 Increasing importance of fuel cost

Rough estimates indicate that the loss of speed due to the drag of the stabilizer fins is approximately ten percent. Recently several papers have discussed a

perfomance criterion for a course autopilot (Koyama, 1967; Norrbin, 1972; Van

Amerongen and Van Nauta Leinke, 1980). It can be shown that the loss of speed is minimized by minimizing the rate of turn of a ship, for instance by applying only small and smooth rudder motions. Powever, the rudder itself causes only a

neglectable small drag. From the data provided by Norrbin, 1972 it follows that, for a cargo liner with 33000 tons displacement and a length of 200 metei-s the loss of speed due to steering is described by (Van Ainerongen and Van Nauta Lemke, 1980)

T

0.0076 2

2

2

( C

+ 1600W +

66

) dt (1)

T

0

The loss of speed caused by the rudder only is thus:

T

0. 0076

) dt % (2)

T

0

A rudder angle of, for instance, 10 degrees gives a loss of speed of nearly five percent, supposed that the ship does not start turning.

It can be shown that for control of the heading high-frequency rudder motions have no positive effect on the course-keeping accuracy (Van Amerongen and Van Nauta Leroke, 1980): course control only necessitates low-frequency rudder motions. With respect to the frequencies of these motions the rolling motion is high frequent. Quick rudder motions, to suppress the rolling motion, with a mean value computed by

other operational cost

repair cod maintenance cost

crew expenses

the course controller, will therefore hardly influence the ship's heading. Because

of eqn.' s (1) and (2) the loss of

speed caused by these

quick rudder ugutlons can be

kept on a reasonable value as long as turning is prevented.

MATHEMATICAL MODELLING

The basic equations which describe the motions of a ship, important with respect to steering and roll stabilization are:

Y = m ( v - ur )

K=I

x

=I

r

z

where m is the ship's mass, included the added mass of the water.

Ix and Iz are the moments of inertia about the x-axis and z-axis.

Y is the hydrodynarnic force in the y-direction.

K and N are hydrodynamic moments.

The other variables have been defined in figure 2a and 2b.

The eqn.'s (3) - (5) can be expanded into a Taylor series. See for instance Eda, 1978. Disregarding all higher order

terms

and introducing the fin angle o yields the following simrlified equations:

Y =Y v

+

Yr

+cp±

S + V r E. K = K v

+ K r +

K

p+

K.'+

K6 + K

c v r

(f

b

N=Nv+Nr++N6+N

o

v

r

C(.

Substitution of eqn.'s () and (6) into (7) and (8), and substitution of eqn. (4) into (7) and eqn (5) into (8) yields, after Laplace transformation:

-3

S222wS +()

r

'4'

c;cc( (10) 'a nt tvp &;co'1 2 (9)

Z0

4z

z In

_{'/ ]}

Disregarding the influence of the fins and the roll angle on the rate of turn, eqn. (10) transforms into the well known Nomoto model. Equations (9) and (10) can be combined into one block diagram as shown In figure 3.

Figure 4 Simplified blockdiagrarn

Figure 3 Blockdiagram of the dynamics between rudder and roll

When the ship has no stabilizing fins and the coupling of and r is disregarded, this blockdiagram simplifies into the system of figure 4. In figure 4 the parameters K and have replaced n and tr . This model will be useful for

nunhr

_6

PARAMETER ESTIMATION FROM FULL-SCALE TRIALS

The simplicity of the model of figure 4 enables to estimate its parameters from full-scale trials. Zig-zag manoeuvres are well stilted as test signals. During the trials the rudder angle, the rate of turn and the roll angle should b recorded. This enables a two-stage identification procedure:

Determine the parameters K and from the rate of turn and rudder signals. N N

Use the rate of turn signal computed by the now identified Nomoto model, together with the rudder and roll angle signals to estimate K , K , z and t)

n

For both stages hillclirnbing with the aid of a digital computer works well. In case that the circumstances are not ideal, for instance when there is wind, it is

necessary to estimate to additional constants r0 and which must be subtracted from the measured r and signals. For obtaining accurate results the constants r0 and should be small.

The parameter-estimation procedure wa tested on data which were available from earlier measurements with a pilot ship. It appeared that for this ship the second order part of the transfer function could he well approximated by one single pole. This yields the block diagram of figure 5.

Figure 5 First order rol dynamics

For this pilot shlp,with a lentgh of 60 meters and sailing with a speed of 12 knots the parameters are given In table i.

The same procedure was used t estimate the parameters of a naval ship, about ttice as long and sailing with a speed of 21 knots. For this ship the parameters of the model of figure 5 have also been determined, but the responses clearly indicated the need of using the second-order roll dynamics of tha model of figure 4. Parameters of both models era given in table 2.

In figures 6 and 7 the measured responses and model responses are given for the pilot ship, (first-order roll dynamics) and for the naval ship (second-order roll dynamics).

15

Figure 6 Results of identification of a pilot vessel

Table 1 paramaters of a pilot vessel

ya dynamics roll dynamics

+ K = 0.125 N = 10 N = 0.4 K

=6

1.9 S p;iqe [-sJ

a

4,

2o°

Figure 7 Results of identification of a naval ship

Table 2 parameters of a naval ship

yaw dynamics K = 0.09 N i:

=6

N I first-order I roll dynamics + = 0.22 K

=5.3

1.9 second-order roll dynamics

+-

-I

K6

0.24

K =5.4

r z 0.23 I

C3,=0.55

In flu(:,!)tr

Figure 8 Bode diagrams for the rudder-heading and the rudder-roll transfer functions.

n a flu r

9

CONCLUS I ONS

It has been shown that relatively simple models can be

derived to describe the transfer between rudder and roll. The parameters of these models can be estimated from full-scale zig-zag manoeuvres. For some ships a model with first-order

roll dynamics appears to give a reasonably good description.

The models also give some insight into the ability of the rudder to stabilizing

a

ship's roil. Due to the non-niinmum phase character of the responses the rudder will never be able to compensate a stationary roll angle, what fins are able to do. Only in the high frequency range the rudder has the desired effect. For low frequencies the roll in opposite direction, caused by the rate of turn will be dominant. However, ala the course control system requires the rate of turn to be kept small.

Figure 8 shows bode diagrams for the transfers between rudder and heading and between rudder and roll, calculated with the second-order roil parameters of table 2. The low-frequency character of the rudder-heading transfer function and the more high-frequency character of the rudder-roll transfer function can clearly be seen.

REFERENCES

J. van Arnerongen and H.R. van Nauta Lemke, "Optimum steering of ships with an adaptive autopilot", Proceedings 5th Ship Control Systems Symposium, Annapolis Md., USA, 1978

J. van Ainerongen and H.P. van Nauta Lemke, "Criteria for optimum steering of

ships", Proceedings Symposium on Ship Steering Automatic Control, Genoa, Italy, 1980

J. van Amerongen, "A model reference adaptive autopilot for ships - Practical results", Proceedings 8th IFAC World Congress, Kyoto, Japan, 1981

A.E. Baitis, "The development and evaluation of a rudder roll stabilization system for the WEEC Hamilton Class", DTNSRPC Report, Bethesda, Nd., USA, 1980

J.B. Carley, "Feasibility study of steering and stabilizing by rudder", Proceedings 4th Ship Control Systems Symposium, The Hague, The Netherlands, 1975

W.E. Cowley and T.H. Lambert, "Sea trials on a roll stabiliser using the ship's rudder", Proceedings 4th Ship Control Systems Symposium, The Hague, The Netherlands, 1975

W.E. Cowley and T.H. Lambert, "The use of the rudder as a roll stabilizer, Proceedings 3rd Ship Control Systems Symposium, Bath, UK, 1972

H. Fda, "A digital simulation study of steering control with effects on roll motions", Proceedings 5th Ship Control Systems Symposium, Annapolis Md., USA, 1978

T. Koyama, "On the Optimum Automatic Steering System of Ships at Sea", J.S.N.A. Japan, Vol. 122, 1967

A.R.J.M. Lloyd, "Roll stabilization by rudder", Proceedings 4th Ship Control Systems Symposium, The Hague, The Netherlands, 1975

S.

Much,

"Hull forms and propulsion plants in the 1980's - Improved fuel economy", Schip en Werf, Vol 47 (26), pp. 443 - 447, 1980

N.P. Norrbin, "On the added resistance due to steering on a straight course", 13th TTTC, Berlin, Hamburg, W. Germany, 1972